What is the number of square units in the area of the hexagon below?

[asy]
unitsize(0.5cm);
defaultpen(linewidth(0.7)+fontsize(10));
dotfactor = 4;
int i,j;
for(i=0;i<=4;++i)

{

for(j=-3;j<=3;++j)

{

dot((i,j));

}

}

for(i=1;i<=4;++i)

{

draw((i,-1/3)--(i,1/3));

}
for(j=1;j<=3;++j)

{

draw((-1/3,j)--(1/3,j));

draw((-1/3,-j)--(1/3,-j));

}

real eps = 0.2;

draw((3,3.5+eps)--(3,3.5-eps));
draw((4,3.5+eps)--(4,3.5-eps));
draw((3,3.5)--(4,3.5));

label("1 unit",(3.5,4));

draw((4.5-eps,2)--(4.5+eps,2));
draw((4.5-eps,3)--(4.5+eps,3));
draw((4.5,2)--(4.5,3));

label("1 unit",(5.2,2.5));

draw((-1,0)--(5,0));
draw((0,-4)--(0,4));
draw((0,0)--(1,3)--(3,3)--(4,0)--(3,-3)--(1,-3)--cycle,linewidth(2));
[/asy]
Each of the four shaded triangles in the diagram below has area $\frac{1}{2}(1)(3)=\frac{3}{2}$ square units, and the shaded triangles along with the hexagon form a rectangular region whose area is $6\cdot4=24$ square units.  Therefore, the area of the hexagon is $24-4\cdot \frac{3}{2}=\boxed{18}$ square units.

[asy]
unitsize(1cm);
defaultpen(linewidth(0.7)+fontsize(10));
dotfactor = 4;

fill((4,0)--(4,3)--(3,3)--cycle,gray);
fill((4,0)--(4,-3)--(3,-3)--cycle,gray);
fill((0,0)--(0,3)--(1,3)--cycle,gray);
fill((0,0)--(0,-3)--(1,-3)--cycle,gray);

int i,j;
for(i=0;i<=4;++i)

{

for(j=-3;j<=3;++j)

{

dot((i,j));

}

}

for(i=1;i<=4;++i)

{

draw((i,-1/3)--(i,1/3));

}
for(j=1;j<=3;++j)

{

draw((-1/3,j)--(1/3,j));

draw((-1/3,-j)--(1/3,-j));

}

real eps = 0.2;

draw((3,3.5+eps)--(3,3.5-eps));
draw((4,3.5+eps)--(4,3.5-eps));
draw((3,3.5)--(4,3.5));

label("1 unit",(3.5,4));

draw((4.5-eps,2)--(4.5+eps,2));
draw((4.5-eps,3)--(4.5+eps,3));
draw((4.5,2)--(4.5,3));

label("1 unit",(5.2,2.5));

draw((-1,0)--(5,0));
draw((0,-4)--(0,4));
draw((0,0)--(1,3)--(3,3)--(4,0)--(3,-3)--(1,-3)--cycle,linewidth(1.5));

[/asy]